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Theorem isnumbasgrplem2 39978
Description: If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)
Assertion
Ref Expression
isnumbasgrplem2 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Proof of Theorem isnumbasgrplem2
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 basfn 16494 . . 3 Base Fn V
2 ssv 3966 . . 3 Grp ⊆ V
3 fvelimab 6719 . . 3 ((Base Fn V ∧ Grp ⊆ V) → ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))))
41, 2, 3mp2an 691 . 2 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
5 harcl 9011 . . . . . 6 (har‘𝑆) ∈ On
6 onenon 9366 . . . . . 6 ((har‘𝑆) ∈ On → (har‘𝑆) ∈ dom card)
75, 6ax-mp 5 . . . . 5 (har‘𝑆) ∈ dom card
8 xpnum 9368 . . . . 5 (((har‘𝑆) ∈ dom card ∧ (har‘𝑆) ∈ dom card) → ((har‘𝑆) × (har‘𝑆)) ∈ dom card)
97, 7, 8mp2an 691 . . . 4 ((har‘𝑆) × (har‘𝑆)) ∈ dom card
10 ssun1 4123 . . . . . . . 8 𝑆 ⊆ (𝑆 ∪ (har‘𝑆))
11 simpr 488 . . . . . . . 8 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
1210, 11sseqtrrid 3995 . . . . . . 7 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
13 fvex 6665 . . . . . . . 8 (Base‘𝑥) ∈ V
1413ssex 5201 . . . . . . 7 (𝑆 ⊆ (Base‘𝑥) → 𝑆 ∈ V)
1512, 14syl 17 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ V)
167a1i 11 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ∈ dom card)
17 simp1l 1194 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑥 ∈ Grp)
18123ad2ant1 1130 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑆 ⊆ (Base‘𝑥))
19 simp2 1134 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑎𝑆)
2018, 19sseldd 3943 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ (Base‘𝑥))
21 ssun2 4124 . . . . . . . . . . 11 (har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆))
2221, 11sseqtrrid 3995 . . . . . . . . . 10 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
23223ad2ant1 1130 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
24 simp3 1135 . . . . . . . . 9 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (har‘𝑆))
2523, 24sseldd 3943 . . . . . . . 8 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (Base‘𝑥))
26 eqid 2822 . . . . . . . . 9 (Base‘𝑥) = (Base‘𝑥)
27 eqid 2822 . . . . . . . . 9 (+g𝑥) = (+g𝑥)
2826, 27grpcl 18102 . . . . . . . 8 ((𝑥 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑐 ∈ (Base‘𝑥)) → (𝑎(+g𝑥)𝑐) ∈ (Base‘𝑥))
2917, 20, 25, 28syl3anc 1368 . . . . . . 7 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (𝑎(+g𝑥)𝑐) ∈ (Base‘𝑥))
30 simp1r 1195 . . . . . . 7 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
3129, 30eleqtrd 2916 . . . . . 6 (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆𝑐 ∈ (har‘𝑆)) → (𝑎(+g𝑥)𝑐) ∈ (𝑆 ∪ (har‘𝑆)))
32 simplll 774 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑥 ∈ Grp)
3322ad2antrr 725 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
34 simprl 770 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (har‘𝑆))
3533, 34sseldd 3943 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (Base‘𝑥))
36 simprr 772 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (har‘𝑆))
3733, 36sseldd 3943 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (Base‘𝑥))
3812ad2antrr 725 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
39 simplr 768 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎𝑆)
4038, 39sseldd 3943 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ (Base‘𝑥))
4126, 27grplcan 18152 . . . . . . 7 ((𝑥 ∈ Grp ∧ (𝑐 ∈ (Base‘𝑥) ∧ 𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥))) → ((𝑎(+g𝑥)𝑐) = (𝑎(+g𝑥)𝑑) ↔ 𝑐 = 𝑑))
4232, 35, 37, 40, 41syl13anc 1369 . . . . . 6 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → ((𝑎(+g𝑥)𝑐) = (𝑎(+g𝑥)𝑑) ↔ 𝑐 = 𝑑))
43 simplll 774 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑥 ∈ Grp)
4412ad2antrr 725 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑆 ⊆ (Base‘𝑥))
45 simprr 772 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑑𝑆)
4644, 45sseldd 3943 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑑 ∈ (Base‘𝑥))
47 simprl 770 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑎𝑆)
4844, 47sseldd 3943 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑎 ∈ (Base‘𝑥))
4922ad2antrr 725 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
50 simplr 768 . . . . . . . 8 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑏 ∈ (har‘𝑆))
5149, 50sseldd 3943 . . . . . . 7 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → 𝑏 ∈ (Base‘𝑥))
5226, 27grprcan 18128 . . . . . . 7 ((𝑥 ∈ Grp ∧ (𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑏 ∈ (Base‘𝑥))) → ((𝑑(+g𝑥)𝑏) = (𝑎(+g𝑥)𝑏) ↔ 𝑑 = 𝑎))
5343, 46, 48, 51, 52syl13anc 1369 . . . . . 6 ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎𝑆𝑑𝑆)) → ((𝑑(+g𝑥)𝑏) = (𝑎(+g𝑥)𝑏) ↔ 𝑑 = 𝑎))
54 harndom 9014 . . . . . . 7 ¬ (har‘𝑆) ≼ 𝑆
5554a1i 11 . . . . . 6 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → ¬ (har‘𝑆) ≼ 𝑆)
5615, 16, 16, 31, 42, 53, 55unxpwdom3 39969 . . . . 5 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆* ((har‘𝑆) × (har‘𝑆)))
57 wdomnumr 9479 . . . . . 6 (((har‘𝑆) × (har‘𝑆)) ∈ dom card → (𝑆* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))))
589, 57ax-mp 5 . . . . 5 (𝑆* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
5956, 58sylib 221 . . . 4 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
60 numdom 9453 . . . 4 ((((har‘𝑆) × (har‘𝑆)) ∈ dom card ∧ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))) → 𝑆 ∈ dom card)
619, 59, 60sylancr 590 . . 3 ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ dom card)
6261rexlimiva 3267 . 2 (∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)) → 𝑆 ∈ dom card)
634, 62sylbi 220 1 ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wrex 3131  Vcvv 3469  cun 3906  wss 3908   class class class wbr 5042   × cxp 5530  dom cdm 5532  cima 5535  Oncon0 6169   Fn wfn 6329  cfv 6334  (class class class)co 7140  cdom 8494  harchar 9008  * cwdom 9016  cardccrd 9352  Basecbs 16474  +gcplusg 16556  Grpcgrp 18094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-omul 8094  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-oi 8962  df-har 9009  df-wdom 9017  df-card 9356  df-acn 9359  df-slot 16478  df-base 16480  df-0g 16706  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-grp 18097  df-minusg 18098
This theorem is referenced by:  isnumbasabl  39980  isnumbasgrp  39981
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